Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.

Calculus is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the lengths, areas and volumes of objects.

Calculus is divided into differential and integral calculus, which are concerned with derivatives

$$\frac{\mathrm{d}y}{\mathrm{d}x}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals

$$\int_a^b f(x)\,\mathrm{d}x = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k,$$

respectively.

134529 questions
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Integrating with Mathway

I wanted to integrate $\int\frac{\sin^4(x)}{\cos(x)}dx$, using “Mathway” and it yielded $\frac{1}{5}\sin^5(x)+C$ by substituting $u=\sin(x)$, My problem is when I do it $u = \sin(x)dx$ $du = \cos(x)dx$ $\frac{du}{\cos(x)} = dx$ I…
Jay Gatsby
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Evaluating $\lim_{x\to 0} \sqrt{x}$

I have a question regarding $$\lim_{x\to 0} \sqrt{x}.$$ Is the limit $0$ or undefined? For the limit to exist both the right hand and left hand limits must exist and be equal. $\lim_{x\to 0^+} \sqrt{x} =0$ but it doesn't even make sense to talk…
Gorg
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Evaluating $ \int_0^1 \frac{x^a - 1}{\ln{}x} dx$

I know how to evaluate using Leibnitz Rule I was trying to see if I could solve it without that $\displaystyle \int_0^1 \frac{x^a - 1}{\ln{}x} dx$ = $\displaystyle \int_0^1 \frac{x^a}{\ln{}x} dx$ - $\displaystyle \int_0^1 \frac{1}{\ln{}x}…
Jinansh Dalal
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Gradient of a real-valued function defined on a non-standard inner product space?

Suppose I have a function $f:\mathbb{R}^n \to \mathbb{R}$. The gradient $\nabla f$ of $f$ at $x$ is the vector in the domain of $f$ satisfying $$[Df(x)] h = \langle \nabla f, h \rangle$$ for all $h$. In this case, it must be that $\nabla f =…
TheProofIsTrivium
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Let $f(x)$ is a piecewise function. when $x$ is not equal to $3$, $f(x)=x^2$, and when $x=3, f(x)=5$. Is it differentiable at the point $x=3$?

Let $f(x)$ is a piecewise function. when $x\ne 3$, $f(x)=x^2$, and when $x=3$, $f(x)=5$. I know that if a function is not continuous, then the function is not differentiable, so $f(x)$ should not be differentiable. But geometrically I think it…
Alex
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How much can you approximate/ignore when taking the limit as n approaches infinity?

Problem: Evaluate the limit as $n$ approaches infinity of $\sqrt[3]{n^3+6n^2+36n+216}-\sqrt[3]{n^3+3n^2+9n+27}.$ If you approximate $n^3+6n^2+36n+216$ as $(n+2)^3$ and $n^3+3n^2+9n+27$ as $(n+1)^3,$ this limit evaluates to $1,$ which is what…
ada
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the difference between discrete sum and integral

hi i have as question we know that that $\frac1x$ is a strictly decreasing function on $[1,\infty[$ let n a natural number greater than 0 let $u_n=\sum_{i=n}^{2n} \frac1i$ and $F(n)=\int_{n}^{2n} \frac1x dx$ i want to prove that $\frac1{2n}\le…
ana nadi lwa3r
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Continuous and non differentiable functions

Can I generate a continuous and non-differentiable function with basic calculus tools? Is there a simple way of expressing such a function?
RamChandra
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Existence of a function where $f(x) > 0, f'(x) < 0, f''(x) < 0$ for all $x\in\mathbb R$

Intuitively, I understand that this is not possible, but I'm trying to provide a concise mathematical explanation. $f(x) > 0$: This means the function is always positive. $f'(x) < 0$: This means the function is always decreasing. $f''(x) < 0$: This…
AlbertB
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Using differentials

I know it's possible to do this: $$\frac{dy}{dx} \frac{dt}{dt} = \frac{dy}{dt} \frac{dt}{dx}$$ but I wonder if this makes sense? $$\frac{d}{dx}\left(\frac{dt}{dt}\right) = \frac{d}{dt} \left(\frac{dt}{dx}\right)$$ so if $t=x^4$ then…
Noteventhetutorknows
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Calculating the first three nonzero terms of the Maclaurin series for f

Let $f$ be a function such that $f^\prime(x)=\sin(x^2)$ and $f(0)=0$.What are the first three nonzero terms of the Maclaurin series for $f\\$ ? I've tried:$f^{\prime\prime}(x)=2x\cos(x^2)$ so $f^{\prime\prime}(0)=0$; …
Aaron Lee
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what is the taylor expansion of $\ln(\sin x/x)$

what is the Taylor expansion of the function $$f(x) = \ln \frac{\sin x}x$$ around the point $x=0$? Ignore powers of $x$ which are greater than $6$. Here is my method: $$\ln(1+x)=x-\frac{x^2}2 + \frac{x^3}3 -\frac{x^4}4,$$ so we should get the…
Iran ehsan alizad
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Extrema of a 2d function

I got to find the extrema of $f(x,y) = (4x^2+y^2)e^{-x^2-4y^2}$ as usual, first derviative and find roots ($e$ already cleaned out): $$8x+(4x^2+y^2)(-2x)=0$$ $$2y+(4x^2+y^2)(-8y)=0$$ But I can't solve that equation for any other than $(x,y)=(0/0)$.…
Reactormonk
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Using Root Test to see whether $\sum_{n=1}^{\infty}\frac{n^{n+\frac{1}{n}}}{(n+\frac{1}{n})^{n}}$ converges

This exercise specifically requires that we use the root test to determine whether the series converges or not. All I've done so far is get the sequence in this form: $$\sqrt[n] \frac{n^{n+\frac{1}{n}}}{(n+\frac{1}{n})^{n}} = \sqrt…
user1162435
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Prove that $\lim\limits_{x\to+\infty}\frac{1}{x}\int_0^x|\sin t|\mathrm{d}t=\frac{2}{\pi}$

I came cross the following equation: $$\lim_{x\to+\infty}\frac{1}{x}\int_0^x|\sin t|\mathrm{d}t=\frac{2}{\pi}$$ I wonder how to prove it. Using the Mathematica I got the following result: Could you suggest some ideas how to prove this? Any hints…
Franz Lee
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